( «9^8 ) 
Example, 
Let the things exposM be a a a b b b c c, or according to 
our way ot notation a» b» c^^ Tis required to find the num- 
ber of their Combinations and Alternations taken 4 and 4. 
Then (becau(e in the things expos'd, there is no one 
thing occurs more than thrice, nor more than three things 
different from each other)will all the forms of Combinati- 
on, which the things expos'd are capable of, be ihefe, 
In the ift form will p=5,q=i, • = ^ = 1, A = j, B = ) 
In the 2d form will p = a, , « = 2, , A = ' 
In the 3d form will p = 2, q = l, « = i, * = 2, A = 3, B =5 
The number of Combinati- u 2 ^ 2 ^ 
ens in the ift form » $ 11 
The number of Combinati- _ A y a — i _ ^JLi — 
©ns in the fecond form a«« — 1 2»i"*^ 
The number of Combina- — A T^^< x-B- ^— f _ 2JLL=r 
tions lathe 3d form « fi» -t 3 » 1 
. And the whole number of Combinations = 10 
Alfo the huraber of Alternations. 
In theift form = 4 " =4 , 4=,« 
.Inthe2dform=3 X - — .. ^ =4.-1 >' =3 »6= 18 
p X n-i X R-^r X n.-j 4^» «>Jli , 
iothe 3d form =3 %T;^^^- ^ =3 -~^i,^^:^3 » 12^=3^ 
And the whole number of Alternations == 70 
Many are the Properties of this Themf^j in common 
with i)thers, as, To find the Vnci4^ of a Multinomial 
rais'd 
